Mathematicians From France

"The duel took place on the 30th in the early morning, and he was grievously wounded by a shot in the abdomen. He was found by a peasant who transported him at 9:30 to the Hôpital Cochin. His younger brother... came and stayed with him, and as he was crying, Evariste tried to console him, saying: "Do not cry. I need all my courage to die at twenty." ...[H]e refused the assistance of a priest. ...[H]e breathed his last ...the following morning."

"When genius... explodes suddenly, at the beginning and not at the end of life, or when we are at a loss to explain its... genesis, we can but feel that we are in the sacred presence of something vastly superior to talent. ...[I]t is not necessary to introduce any mystical idea, but it is one's duty to acknowledge the mystery. ...Galois' fateful existence helps one to understand Lowell's saying: "Talent is that which is in a man's power, genius is that in whose power man is." If Galois had been simply a mathematician of considerable ability, his life would have been far less tragic, for he could have used his mathematical talent for his own advancement and happiness; instead... the furor of mathematics—as one of his teachers said—possessed him and he had no alternative but absolute surrender to his destiny."

"It is painful to think that a few rays of generosity from the heart of his elders might have saved this boy or... sweetened his life."

"Galois... accomplished his task and... few men will... accomplish more. He has conquered the purest kind of immortality. As he wrote to his friends: "I take with me to the grave a conscience free from lie, free from patriot's blood". How many of the conventional heroes of history, how many of the kings, captains and statesmen could say the same?"

"Although the Greek use of s in the case of greatest common measure was well known in the Middle Ages, the modern theory of the subject may be said to have begun with Bombelli (1572). ...The first great memoir on the subject was Euler's De fractionibus continuis (1737), and in this work the foundation for the modern theory was laid. Among other interesting cases Euler developed e as a continued fraction, thus: e = 2 + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{4 + \cfrac{1}{1+...}}}}} Of the later contributors to the theory, special mention should be made of Lagrange (1767) and Galois."

"The modern in general is commonly said to date from Abel and Galois. The latter's posthumous (1846) memoir... established the theory... To him is due the discovery that to each equation there corresponds a group of substitutions (the "group of the equation") in which are reflected its essential characteristics. Galois' early death left without sufficient demonstration several important propositions, a gap which has since been filled."

"The first formulas for the computation of the s of the roots of an equation seem to have been worked out by Newton, although Girard (1629) had given, without proof, a formula for a power of the sum, and Cardan (1545) had made a slight beginning in the theory. In the 18th century Lagrange (1768) and Waring (1770, 1782) made several valuable contributions to the subject, but the first tables, reaching to the tenth degree, appeared in 1809 in the Meyer Hirsch Aufgabensammlung. In Cauchy's celebrated memoir on determinants (1812) the subject began to assume new prominence, and both he and Gauss (1816) made numerous and important additions to the theory. It is, however, since the discoveries by Galois that the subject has become one of great significance."

"Gauss took as the subject for his doctor’s dissertation (1799) a proof of the : an algebraic equation has a root of the form a + bi, a , b real. ...After Girard’s conjecture, there had been attempts at proof, including essays by D’Alembert (1746) and Euler (1749). All were faulty, as were the first and fourth attempts (1799) by Gauss. ...[T]he fundamental theorem in its classic form, as proved in the theory of functions of a complex variable, is no longer regarded as belonging to algebra. It is supplanted in modern algebra by a statement which is almost a triviality. The basic ideas of the modern treatment go back to Galois.., Dedekind.., and Kronecker.., not to Gauss."

"Three new approaches to number, in 1801 and in the 1830's, were to hint at the general concept of mathematical structure and reveal unsuspected horizons... That of 1801 was the concept of congruence, introduced by Gauss [age 24] in... the Disquisitiones arithmeticae... To this and the revolutionary work (1830-2) of E. Galois... in the theory of algebraic equations can be traced the partial execution of L. Kronecker’s... revolutionary program in the 1880's for basing all mathematics on the s."

"We must... indicate the involuntary participation of E. Galois... and N. H. Abel... in the development of Kronecker’s Pythagoreanism. Galois... adhered to no such creed. Nor did Abel. But it was in the attempt to understand and elucidate the of equations, left (1832)... in a... fragmentary and unapproachable condition, that Kronecker acquired some of his skill. Both Kronecker and Dedekind, two of the founders (E. E. Kummer... being a third) of the theory of algebraic numbers, were inspired partly by their scrutiny of the Galois theory to begin their... revolutionary work in algebra and arithmetic. ...Galois and Abel mark the beginning of one modern approach to algebra. The transition from highly finished individual theorems to abstract and widely inclusive theories..."

"Younger than Gauss by thirty-four years, and dying twenty-three years before him, Galois... seems more modern... Gauss terminated his investigations on the nature of the solutions of with the binomial equationGalois grasped and solved (1830) the general problem, proving, among other things, necessary and sufficient conditions for the solution by radicals of any algebraic equation. Mathematics after Gauss, and partly during his own lifetime, became more general and more abstract... Interest in special problems sharply declined if there was a general problem including the special instances to be attacked... [i.e.,] mathematics after Gauss turned to the construction of inclusive theories and general methods which, theoretically... implied... detailed solutions of infinities of special problems. In this sense Galois was more modern..."

"Applications, or developments, of... extensions of number followed two main directions. ...The second, in the arithmetical spirit of Gauss, guided in part by the abstract algebraic outlook of Galois, led to a partial but extensive arithmetization of algebra."

"Of all these influences, two in particular are germane... the development of ; and the infiltration of the ideas of Abel and Galois into algebra as a whole. The of equations was acknowledged by both Dedekind and Kronecker to be the inspiration for their own general and semi-arithmetical approach to algebra. Two of the basic concepts of the Galois theory, domains of rationality, or fields, and groups, were the point of departure. ...Common algebra is the most familiar example of a field."

"The earliest recognitions of fields, but without explicit definition, appear to be in the researches of Abel (1828) and Galois (1830-1) on the solution of equations by radicals. The first formal lectures on the were those of Dedekind to two students in the early 1850's. Kronecker also at that time began his studies on abelian equations. It appears that the concept of a field passed into mathematics through the arithmetical works of Dedekind and Kronecker."

"There are six major episodes to be observed, four of which will be described... The four are the definition by Gauss, E. E. Kummer.., and Dedekind of s; the restoration of the in s by Dedekind’s introduction of ideals; the definitive work of Galois on the solution of algebraic equations by radicals, and the theory of finite groups and the modern theory of fields that followed; the partial application of arithmetical concepts to certain linear algebras by R. Lipschitz.., A. Hurwitz.., L. E. Dickson.., Emmy Noether.., and others. All of these developments are closely interrelated."

"Galois... made the terminal contribution so far as are concerned, and subsequent reworkings of his initial theory have added nothing basically new to his criteria for solvability by radicals. Even the modernized presentation of the , as in the streamlined model of E. Artin... is a tribute to the mathematical creed of Galois, in its elimination of all superfluous machinery. For this modern release from algebraic calculation, the direct approach of A. E. ‘Emmy’ Noether... in the 1920’s was primarily responsible. Much of her mathematics was in the spirit of Galois. But his methods, sharpened and generalized by his successors, have transcended the problem for which they were invented, and have rejuvenated much of living pure mathematics."

"[C]oncerning the vital residue of the theory of algebraic numbers... it, like the , can be traced to definite, highly special problems. Neither Galois nor the creators of the theory of s set out deliberately to revolutionize a mathematical technique; their comprehensive methods were invented to solve specific problems. Such appears to have been the usual path to abstractness, generality, and increased power. Some difficult problem... is taken as the point of departure without any conscious effort to create a comprehensive theory; repeated failures to achieve a solution by known procedures force the invention of new methods; and finally, the new methods, having been necessitated by a problem which appeared in the historical development, themselves pass into the main stream."

"The Galois theory of equations itself was the concluding episode in about three centuries of effort to penetrate the arithmetical nature of the roots of algebraic equations."

"Lagrange did not explicitly recognize groups. Nevertheless, he obtained equivalents for some of the simpler properties of s. For example, one of his results, in modern terminology, states that the order of a of a divides the order of the group. Normal (self-conjugate, invariant) subgroups, basic in the theory of algebraic equations and in that of group structure, were introduced by Galois, who also invented the term 'group.'"

"Both Abel and Galois were indebted to Lagrange in their own, profounder work on s. ...The unique importance of Abel’s proof is that it inspired Galois to seek a deeper source of solvability, which he found in the theorem that an is solvable by radicals its group, for the field of its s, is solvable."

"The simple of any two groups was defined in connection with the postulates for a group. Galois considered simply isomorphic groups as the same group which, abstractly, they are."

"The earliest discussion from the standpoint of groups of the (modular) equations arising in the division of s was by Galois."

"The work of Galois and his successors showed that the nature, or explicit definition, of the roots of an algebraic equation is reflected in the structure of the group of the equation for the field of its coefficients. This group can be determined nontentatively in a finite number of steps, although, as Galois himself emphasized, his theory is not intended to be a practical method for solving equations. But, as stated by Hilbert, the Galois theory and the theory of algebraic numbers have their common root in that of algebraic fields. The last was initiated by Galois, developed by Dedekind and Kronecker in the mid-nineteenth century, refined and extended in the late nineteenth century by Hilbert and others, and finally, in the twentieth century, given a new direction by the work of Steinitz in 1910, and in that of E. Noether and her school since 1920."

"Even before Galois coined the term 'group,' A. L. Cauchy... made (1815) extensive investigations in what are now called s, and discovered some of the simpler basic theorems."

"[A]ny mathematician today must be impressed by the apparent permanence of the ideas introduced by Abel... and Galois.., and the profound difference between their approach to mathematics and that of their predecessors including, in some respects, Gauss... To these young men, perhaps more than to any other two mathematicians, can be traced the pursuit of generality which distinguishes the mathematics of the recent period, beginning with Gauss in 1801, from that of the middle period. They initiated... the deliberate search for inclusive methods and comprehensive theories. Their forerunners in the middle period were Descartes with his general method in geometry; Newton and Leibniz with the differential and integral calculus created to attack the mathematics of continuity by a uniform procedure; and Lagrange, with his universal method in mechanics. Their contemporary in recent mathematics was Gauss, who in his arithmetic sought to unify much of the uncorrelated work of the leading arithmeticians from Fermat to Euler, Lagrange, and Legendre. Both Abel and Galois acknowledged their indebtedness to the theory of cyclotomy created by Gauss; and although they went far beyond him in their own algebra (Abel in analysis also), it is at least conceivable that neither Abel nor Galois would have chosen the road he followed had it not been for the hints in the Gaussian theory of binomial equations."

"Both Abel and Galois died long before their time, Abel at the age of twenty-seven from tuberculosis induced by poverty, Galois at twenty-one of a pistol shot received in a meaningless duel. When Abel’s genius was recognized, he was subsidized by friends and the Norwegian government. By nature he was genial and optimistic. Galois spent a considerable part of his five or six productive years in a hopeless fight against the stupidities and malicious jealousy of teachers and the smug indifference of academicians. Not at first quarrelsome or perverse, he became both."

"Whoever, if anybody, was responsible for the colossal waste represented by these two premature deaths, it seems probable that mathematics was needlessly deprived of the natural successsors of Gauss. What Abel and Galois might have accomplished in a normal lifetime cannot be even conjectured. ...Early maturity and sustained productivity are the rule, not the exception, for the greatest mathematicians. It may be true that the most original ideas come early; but it takes time to work them out."

"Gauss, the penniless son of a day laborer, was educated by society as represented by the Duke of Brunswick. Today he would be educated at public expense... An Abel, no doubt, would be sent by the municipal health authorities to a sanitarium, where he might recover. A Galois... would find himself at outs with respectability, or in the protective custody of the police on some trumped-up charge... or in a concentration camp. For there is but little evidence that teachers are less helpless in the disturbing presence of a mind of the very highest intelligence than they were... or that the guardians of law and order are less nervous than they were when they sentenced Galois to... jail on a legal technicality. Aesop’s fable of the peacock and the crows has an element of permanence... you are different from us; get out or be plucked."

"Congruences were responsible for one theory of far more than merely arithmetical interest. The notation for a congruence suggests the introduction of appropriate 'imaginaries' to supply the congruence with roots equal in number to the degree of the congruence when there is a deficiency of real roots. As in the corresponding algebraic problem, it is not obvious that imaginaries can be introduced consistently. That they can, was first proved in 1830 by Galois, who invented the required 'numbers,' since called Galois imaginaries, for the solution of any irreducible congruence F(x) = 0 mod p , where p is prime. He thus obtained a generalization of Fermat’s theorem, and laid the foundation of the theory of s. ...Galois was eighteen when he invented his imaginaries."

"Probably almost anyone who has ever seriously attempted to solve differential equations by the will appreciate the labor inherent in any such heroic project as Wilczynski’s and agree with Galois that, whatever the nature of its unchallenged merits, the theory of groups does not afford a practicable method for solving equations. Galois of course was speaking of s, but his opinion, in the judgment of experts in the Lie theory, carries over to differential equations. Beyond a not very advanced stage of complexity, the calculations become prohibitive to even the most persevering obstinacy."

"[A]s noted by Lie in the grand summary (1893) of his lifework, the nineteenth century’s greatest effort in formal algebra—as distinguished from the more abstract, structural algebra originating with Galois—was absorbed in analysis."

"Galois, who was Lie's idol, indirectly inspired the application of continuous groups to differential equations. In a letter of 1874 to A. Mayer, Lie observes that "In the theory of algebraic equations before Galois only these questions were proposed: Is an equation solvable by radicals, and how is it to be solved? Since Galois, among other questions proposed is this: How is an equation to be solved by radicals in the simplest way possible? I believe the time is come to make a similar progress in differential equations.""

"Just as the algebraic theory characterizes the nature of the irrationalities required for the solution of a given algebraic equation, so does the structural theory of differential equations characterize and classify the functions defined by a system of differential equations. ...[t]he initial impulse for a structural theory came from Lie's transformation groups. In spite of Picard's deprecatory estimate of his own contribution, which, historically, inaugurated the project, as "only a very natural extension to an analytic problem of the extremely fruitful ideas introduced into algebra by Galois," the problem of devising a structural theory for differential equations was no facile exercise in principles already classic."

"Although... structural theories of a major division of analysis originated in the late nineteenth century, they are more in the spirit of the general analysis of the twentieth. Their primary objectives are to discover what can be done rather than to do it, and to give criteria for what cannot be done. ...[A]s in Abel’s proof that the general quintic is not solvable by radicals, a demonstration of impossibility definitely disposed of what might seem a reasonable problem. Once more the methodology of Abel and Galois made an outstanding contribution to the development of mathematics. In this connection it is interesting to recall Lie’s opinion that the pattern of nineteenth century mathematics was laid out by four men, Gauss, Cauchy, Abel, and Galois."

"With this highly abstract definition of a space in mind, we return once more to Klein’s program and its successors. It is interesting to observe the abstract identity between the following description of spacial structure and structure as described in connection with modern algebra, and further to note once more that the basic concepts originated with Galois. Two spaces are called equivalent or (simply) isomorphic if there is a one-one correspondence between the objects in the spaces which establishes a one-one correspondence between all the properties constituting the structures of the respective spaces. When this is applied to two spaces which are the same, there is thus defined what is called an of the space. It follows... from these definitions that all the automorphisms of a given space form a group."

"Galois... had nothing of the topologist about him..."

"It is only mental habit that prevents us from realizing how miraculous it is that motion can be passed from one body to another. Once our eyes have opened, nothing is so striking. For those who have never thought about it, it doesn't seem mysterious; by contrast, those who have meditated on it may despair of ever understanding it."

"Maupertuis really had no principle, properly speaking, but only a vague formula, which was forced to do duty as the expression of different familiar phenomena not really brought under one conception. ...Maupertuis' performance, though it had been unfavorably criticized by all mathematicians, is, nevertheless, sort of invested with a sort of historical halo. It would seem almost as if something of the pious faith of the church had crept into mechanics. However, the mere endeavor to gain a more extensive view... was not altogether without results. Euler, at least, if not also Gauss, was stimulated by the attempt of Maupertuis."

"After having worked in the theory of light and gravitation, he announced, in 1744, a new minimum principle, the Principle of Least Action, from which he claimed he could deduce the behavior of light and masses in motion. The principle asserts that nature always behaves so as to minimize an integral known technically as action, and amounting to the integral of the product of mass, velocity, and distance traversed by a moving object. From this principle he deduced the Newtonian laws of motion. With sometimes suitable and sometimes questionable interpretation of the quantities involved, Maupertuis managed to show that optical phenomena, too, could be deduced from this principle. Hence, to an extent at least, he succeeded in uniting the optics of the eighteenth century and mechanical phenomena. ... Maupertuis advocated his principle for theological reasons. ...He ...proclaimed his principle to be not only a universal law of nature but also the first scientific proof of the existence of God, for it was "so wise a principle as to be worthy only of a Supreme Being."

"According to Du Bois Reymond, Maupertuis's teleological tendencies showed themselves early in his career in speculations as to what grounds the Creator could have had for preferring the law of the inverse square to all other possible laws of attraction. ... Maupertuis read to the Paris Academy on the 20th of February, 1740, a memoir entitled: "Loi du Repos des Corps." He began by remarking that demonstrations a priori of such principles as that of the conservation of vis viva "cannot apparently be given by physics; they seem to belong to some higher science." ... Maupertuis's first enunciation of the law of the least quantity of action was in a memoir read to the French Academy on April 15th, 1744, entitled "Accord de différentes Loix de la Nature qui avoient jusqu'ici paru incompatibles." The laws in question appear to be those of the reflection and of the refraction of light. When a ray of light in a uniform medium travels from one point to another, either without meeting an obstacle or with meeting a reflecting surface, nature leads it by the shortest path and in the shortest time. But when a ray is refracted by passing from a uniform medium to one of different density, the ray neither describes the shortest space nor does it take the shortest time about it. As Fermat showed, the time would be the shortest if light moved more quickly in rarer media, but Newton proved that, as Descartes had believed, light moves more quickly in denser media. Maupertuis's discovery was that light neither takes always the shortest path nor always that path which it describes in the shortest time, but "that for which the quantity of action is the least.""

"The quantity of action is the product of the mass of the bodies times their speed and the distance they travel. When a body is transported from one place to another, the action is proportional to the mass of the body, to its speed and to the distance over which it is transported."

"When a change occurs in Nature, the quantity of action necessary for that change is as small as possible."

"After so many great men have worked on this subject, I almost do not dare to say that I have discovered the universal principle upon which all these laws are based, a principle that covers both elastic and inelastic collisions and describes the motion and equilibrium of all material bodies. This is the principle of least action, a principle so wise and so worthy of the supreme Being, and intrinsic to all natural phenomena; one observes it at work not only in every change, but also in every constancy that Nature exhibits. In the collision of bodies, motion is distributed such that the quantity of action is as small as possible, given that the collision occurs. At equilibrium, the bodies are arranged such that, if they were to undergo a small movement, the quantity of action would be smallest. The laws of motion and equilibrium derived from this principle are exactly those observed in Nature. We may admire the applications of this principle in all phenomena: the movement of animals, the growth of plants, the revolutions of the planets, all are consequences of this principle. The spectacle of the universe seems all the more grand and beautiful and worthy of its Author, when one considers that it is all derived from a small number of laws laid down most wisely. Only thus can we gain a fitting idea of the power and wisdom of the supreme Being, not from some small part of creation for which we know neither the construction, usage, nor its relationship to other parts. What satisfaction for the human spirit in contemplating these laws of motion and equilibrium for all bodies in the universe, and in finding within them proof of the existence of Him who governs the universe!"

"The elements that make up all other bodies, these must be bodies that are perfectly inelastic, undeformable and unchangeable."

"A true philosopher does not engage in vain disputes about the nature of motion; rather, he wishes to know the laws by which it is distributed, conserved or destroyed, knowing that such laws is the basis for all natural philosophy."

"Research into motion was not to the liking (or perhaps not within the scope) of the ancients, so that we may consider it as a completely new science. How could the ancients have discovered the laws of moiton, given that some philosophers reduced all their speculations about motion to sophistic disputes, whereas others denied that motion existed at all?"

"One should not be deceived by philosophical works that pretend to be mathematical, but are merely dubious and murky metaphysics. Just because a philosopher can recite the words lemma, theorem and corollary doesn't mean that his work has the certainty of mathematics. That certainty does not derive from big words, or even from the method used by geometers, but rather from the utter simplicity of the objects considered by mathematics."

"Despite the disorder observed in Nature, one finds enough traces of the wisdom and power of its Author that one cannot fail to recognize Him."

"It is interesting to note that Newton was not impressed by Descartes' great argument for God's existence derived from the idea of a perfect Being, nor by other metaphysical arguments that we have mentioned; yet Newton's own arguments for God's existence from the uniformity and suitability of different parts of the universe would not have seemed like proofs to Descartes."

"On the 15th of April 1744, I described the principle upon which the following work is based, in the public assembly of the Royal Academy of Sciences of Paris, as reported in the Acts of that academy. At the end of the same year, Professor Euler published his excellent book Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. In a supplement to his book, this illustrious geometer showed that, in the trajectory of a particle acted on by a central force, the velocity multiplied by the line element of the trajectory is minimized. This observation gave me great pleasure, as a beautiful application of my principle to the motion of the planets, which is determined by this principle. From the same principle, I will now try to derive higher and more important truths."